ERIC EJ807392: Teaching Mathematical Problem Solving to Middle School Students in Math, Technology Education, and Special Education Classrooms

2004·0.25 MB·English

by  ERIC

RMLE Online—Volume 27, No. 1
RMLE
Online
Research in
Middle Level Education
David Hough, Ph.D., Editor
Missouri State University
Springfield, Missouri
2004 • Volume 27 (cid:129) Number 1 ISSN 1084-8959
Teaching Mathematical Problem Solving to Middle School Students in Math,
Technology Education, and Special Education Classrooms
Brian A. Bottge
Mary Heinrichs
Zara Dee Mehta
Enrique Rueda
Ya-Hui Hung
Jeanne Danneker
University of Wisconsin-Madison
Citation: Bottge, B. A., Heinrichs, M., Mehta, Z. D., Rueda, E., Hung, Y., & Danneker, J. (2004). Teaching mathe-
matical problem solving to middle school students in math, technology education, and special education class-
rooms. Research in Middle Level Education Online, 27(1). Retrieved [date], from
http://www.nmsa.org/Publications/RMLEOnline/tabid/101/Default.aspx
Abstract
This study compared two approaches for teaching sixth-grade middle school students to solve math problems
in math, technology education, and special education classrooms. A total of 17 students with disabilities and
76 students without disabilities were taught using either enhanced anchored instruction (EAI) or text-based
instruction coupled with applied problems (TBI). Results showed that both EAI and TBI students benefited
from instruction in their math class, but EAI students were able to maintain and transfer what they learned in
the technology education classroom several weeks later. The performance level of students with disabilities
was low in both groups, but additional small-group instruction in special education settings helped several stu-
dents with disabilities achieve at levels commensurate with their peers without disabilities.
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In its recent report, the National Middle School Association (NMSA) emphasized the importance of integrat-
ing curriculum formats to help middle school students master basic skills and develop more complex levels of
thinking (NMSA Position Statement on Curriculum Integration, NMSA, 2002). The authors suggest that teams
of teachers should identify common concepts in their curricula that are grounded in real-world contexts and
then teach them across subject areas. Accomplishing this goal would help to ensure that every student is pre-
pared for life beyond school, a theme emphasized in theCarnegie Corporation Report, Turning Points 2000:
Educating Adolescents in the 21st Century (Carnegie Corporation, 2000).
The NMSA and Carnegie recommendations are closely aligned with workforce expectations. For example, gov-
ernment reports (e.g., What Work Requires of Schools: A SCANS Report for America 2000, U.S. Department of
Labor, 1991) state that success in the workplace depends on the ability of workers to use their academic skills
in social contexts to solve important problems. In addition to these content-oriented skills, employees are
expected to communicate orally and in writing with individuals of diverse backgrounds, form and test hypothe-
ses, and perform simple tasks on personal computers (Gray & Herr, 1998; Murnane & Levy, 1996).
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Helping students identify and understand the contextual elements of problem situations is not a new idea.
Early in the last century, several writers recognized the importance of context in transferring skills from one
situation to another (Thorndike, 1922; Thorndike & Woodworth, 1901; Whitehead, 1929; Woodworth, 1921).
Contemporary theorists have also emphasized the importance of incorporating contextual factors in learning,
naming this pedagogical approach situated cognition (Brown, Collins, & Duguid, 1989; Greeno & the Middle
School Mathematics Through Applications Project Group, 1998; Lave, Smith, & Butler,1988) or contextual-
izedor anchored instruction(Bransford, Sherwood, Hasselbring, Kinzer, & Williams, 1990; Cognition and
Technology Group at Vanderbilt, 1997).
Despite the progress in mathematics reform (e.g., National Council of Teachers of Mathematics, 1989, 2000),
large differences still exist in the ways students use mathematics in school and workers apply mathematics in
job settings (Boaler, 1998; Masingila, 1993; Smith, 1999). School-based problems are typically embedded in
text-laden descriptions, which some students with low math andreading skills cannot understand and do not
find motivating (Cawley, Parmar, Foley, Salmon, & Roy, 2001; Ellis, 1998; Miller & Mercer, 1997; Mtetwa &
Garofalo, 1989). The inability to comprehend text severely limits these students' ability to construct the men-
tal models, or pictures, of problem contexts that are necessary for solving authentic problems (Norman, 1983;
Shepard & Cooper, 1982).
In mathematics, essential work skills include the ability to compute fractions and decimals, interpret tables
and graphs, manipulate algebraic formulas, and solve problems with fellow workers. Ethnographic studies
have provided insight into the kinds of mathematics problems that workers solve on the job. For example, car-
pet layers know that carpet comes in standard widths, the nap must all run the same way, and seams should be
placed out of heavy traffic areas (Masingila, 1993). Tile installers need to be able to lay tile symmetrically
from the center of a room and then shift tiles as needed to eliminate or reduce waste at the outer edges.
Automobile assembly workers measure tolerances in English or metric systems according to machine and part
specifications (Murnane & Levy, 1996; Smith, 1999). In each profession, the problems and procedures (i.e.,
number and nature of math operations) are constrained by the context.
For several years, we have investigated ways to help students with and without learning disabilities gain deep-
er understandings of mathematics in contextualized learning environments based on a theoretical model of
teaching and learning mathematics we call the Key Model (Bottge, 2001) (see Figure 1).
At the center of the model is Enhanced Anchored Instruction (EAI), a pedagogical approach that contrasts sharply
with traditional text-based instruction (TBI). EAI is a form of anchored instruction (Cognition and Technology
Group at Vanderbilt, 1997) that situates, or "anchors," problems in authentic contexts (Newmann, Secada, &
Wehlage, 1995) that students find meaningful. Video-based problems on CDs, called anchors, are especially moti-
vating for students with low reading skills because they can immediately access the problems embedded in the
anchor without having to decode and comprehend word-based problems. EAI also extends students' learning by
affording them opportunities to apply their skills in building hands-on projects (e.g., skateboard ramps, compost
bins, hovercrafts). EAI brings together teams of mathematics, special education, and technology education teach-
ers with diverse but complementary skills to plan, develop, and deliver the unique curriculum.
Previous studies have shown that average- and low-achieving students who were taught with EAI solved
sophisticated problems in mainstream and remedial math classrooms and were more successful in applying
what they learned in technology education classrooms than students who were taught with TBI (Bottge, 1999;
Bottge & Hasselbring, 1993; Bottge, Heinrichs, Chan, & Serlin, 2001). In a recent study conducted in inclu-
sive classrooms, the same positive results were found for students without disabilities (Bottge, Heinrichs,
Mehta, & Hung, 2002). However, the findings were equivocal for students with disabilities. Close observa-
tions of students with learning disabilities showed that some either copied the work of more capable students
or did not participate at all.
The overall purpose of this study was to compare students' math performance after they learned with EAI or
TBI in their math classroom and to assess their problem-solving skills in the technology education classroom
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several weeks later. The second objective was to investigate how much additional instruction was necessary to
help students with disabilities perform at the level of their average-achieving classmates.
Method
Participants
A total of 93 sixth-grade students in four math classes in a middle school located in the upper Midwest partic-
ipated in the study, which spanned seven months from November through May. Descriptive information is
provided in Table 1. Seventeen students were receiving special education services and 13 of these students had
been diagnosed with LD. According to the Wisconsin Department of Public Instruction (2002), students may
have a learning disability when they do not achieve commensurate with their age and ability levels and have
functional achievement at or below 50% of expected achievement (derived by the formula: IQ x Years in
School x .5 = Grade Score).
Two math teachers (Ann and Marj) each taught one EAI class and one TBI class during the first phase of the
study. Ann had a master’s degree in learning disabilities and reading and an additional 24 credits in curricu-
lum. She was in her 26th year of teaching and was licensed to teach general education in Grades 1–8, reading
in Grades K–12, and all students with disabilities except those with sight and hearing impairments. The other
math teacher, Marj, had taught general education for nine years in Grades 1–6 and was also licensed to teach
students with learning disabilities (LD) in Grades K–8. Connie, a special education teacher, had a master’s
degree in emotional disabilities (ED), certification to teach LD, and was in her sixth year of teaching. She
assisted students with LD/ED in the general education classrooms and also helped students who needed addi-
tional academic assistance in her resource room. Russ was in his second year of teaching and taught the tech-
nology education classes.
Classes of students were randomly assigned to EAI or TBI because the school schedule could not be changed
to assign individual students to instructional groups. Non-White students constituted a higher percentage of
Marj’s classes (18/47 or 38%) than Ann’s classes (13/46 or 28%), although the overall percentage of non-
White EAI and TBI students was equal (33%). Marj’s classes also included a larger percentage of students
with disabilities (28%) than Ann’s classes (9%), but the percentage of students with disabilities in EAI and
TBI was almost the same (18% and 19%, respectively), as was the total number of minutes per week students
received special education services.
On pretests administered prior to instruction, the means and standard deviations of all EAI and TBI students
were 26.1 (SD = 11.0) and 25.3 (SD = 11.8) on fractions computation, t(89) = .33, p= .74, 24.0 (SD = 10.1)
and 23.7 (SD = 9.7) on the word problem test, t(88) = .16, p= .88, and 6.1 (SD = 3.2) and 5.4 (SD = 3.7) on
the video problem test, t(78) = .89, p= .38. An attitude questionnaire was also administered to gauge the stu-
dents’general impressions about math. In both groups, just over a third of the students indicated they like
math.
Research Design and Instrumentation
We employed a nonequivalent pretest-posttest design coupled with a finer-grained analysis of teacher and stu-
dent performance to answer the research questions. Several authors (e.g., Gersten, Baker, & Lloyd, 2000;
Mastropieri, Scruggs, & Shiah, 1991; Woodward & Baxter, 1997) have urged researchers who study special
populations to use experimental methods to test the effects of interventions on student achievement and in-
depth descriptions to uncover the contextual factors that may have contributed to these performances.
The study was conducted in two phases. During Phase 1, Ann’s and Marj’s mathematics classes were random-
ly assigned to either EAI or TBI, yielding two EAI and two TBI groups. Immediately before and after instruc-
tion, students in the four classrooms took a test emphasizing computation with fractions and two problem-
solving tests. A week after students finished the posttests, three students with LD and one student with the
disability label, Other Health Impaired (OHI), (2 from EAI and 2 from TBI) worked in pairs in the special
education resource room to learn the concepts that they had not learned, as documented by their scores on the
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problem-solving posttests. Connie, who helped the students with disabilities in the mathematics classrooms,
supervised the work of the students and answered questions to clarify obvious misconceptions they had about
the problems. Immediately after they had solved the EAI video-based problem, they took the video problem-
solving test again.
Phase 2 took place in technology education classes that included cohorts of students (including students with
disabilities) who learned with EAI or TBI in their math classes 6, 13, and 22 weeks before. The main objec-
tive of the class was to engage students in applying the skills they had learned in the math class to plan and
build a hovercraft frame in the technology education classroom. Students were assessed before and after they
worked on the project.
The content validity of the tests was supported by their use in previous studies (e.g., Bottge, 1999; Bottge &
Hasselbring, 1993; Bottge et al., 2002) and by previous research showing that the test items reliably estimated
mathematics achievement (Cohen, Bottge, & Wells, 2001). We calculated the interrater reliability of each
measure on 20% of the test protocols from a randomly selected sample of pretests and posttests by dividing
the number of agreements by the total number of agreements and disagreements and multiplying by 100
(Sulzer-Azaroff & Mayer, 1977).
Fractions computation test. An 18-item, 36-point computation test assessed students’ability to compute frac-
tions. Twelve items asked students to add and subtract simple fractions with like denominators, mixed num-
bers without renaming, and mixed numbers with renaming. We awarded students one point if they correctly
renamed fractions before computing them and one point if they expressed the correct answer in simplest
terms. Cronbach’s coefficient alpha was .98, and interrater reliability was 99% (range = 94%–100%).
Word problem test. We developed an 18-item, 36-point test with a reading level at or lower than fourth grade
to measure students’ability to solve single-step and multistep word problems. Students could earn one point
per item for showing how to solve the problem (i.e., correct operations) and one point per item for arriving at
the correct answer. The content of the items paralleled the instruction in EAI and TBI, which included using
information from charts and figures to compute linear measurements and calculating construction costs.
Cronbach’s coefficient alpha was .97, and interrater reliability was 99% (range = 92%–100%).
Video problem test. Based on work from previous studies (Bottge et al., 2002), we developed a 36-point
assessment to measure students’ability to solve the problem presented in the video Fraction of the Cost. The
test was finalized only after undergoing five revisions based on the recommendations of two raters who scored
protocols of students not involved in the study. Partial or full credit was awarded on 15 procedures in five
major categories: computing money; indicating the lengths of two-by-fours (2 x 4 dimensional lumber) the
students measured in the garage; converting lengths of wood from feet to inches; showing combinations of
lengths of lumber; and calculating the total cost.
The test was administered in math class in the following way: First, the teacher asked students to watch the
video once without interruption and to pay close attention to the questions at the end. After checking the stu-
dents’understanding of the problems they were to solve, the teacher showed the video a second time. Then
the teacher gave each student a five-page packet that contained relevant and irrelevant information for solving
the problems in the video, such as a savings statement from a bank, building plans for a skateboard ramp, and
a newspaper flier showing a sale on building supplies. The teacher asked the students to show how to solve
the problems in a 6 x 6 in. work area on the problem response form. Cronbach’s coefficient alpha was .80,
and interrater agreement was 94% (range = 64%–100%).
Hovercraft problem test. A 22-point performance-based assessment in the technology education class asked
students to show how they would use lengths of polyvinyl chloride (PVC) pipe in the most economical way to
build a “rollover cage” for a hovercraft, based on a drawing, a materials list, and a specified amount of money.
The content of the problem closely aligned with the problems students had solved in EAI and TBI in their
math class. Students had to calculate and compare sums of money, add and subtract fractions, and figure out
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combinations of PVC pipe that would enable them to build the hovercraft cage with the money available.
Unlike the problems they solved in math class, the hovercraft problem required students to make their calcula-
tions based on real-life constraints. For example, PVC pipe is sold only in 10 ft. lengths, unlike two-by-fours,
which are available in 6 ft., 8 ft., and 10 ft. lengths. Students also had to figure out how many 45- and 90-
degree connectors they needed to hold the cage together, calculate the cost of the connectors, and add this
amount to the total cost.
Like the problems in TBI, the hovercraft problem included diagrams and text-based descriptions of the prob-
lem on paper (not video). Like the problems in EAI, the text-based descriptions were brief and easy to read.
Students described their solutions in an open work area on a single page of paper as they had on the problem-
solving tests. The scoring protocols included eight criteria we had identified from field tests and previous
studies. Raters were trained on scoring papers with the protocols from tests administered to students who
were not involved in this study. Cronbach’s coefficient alpha was .94, and interrater agreement was 91%
(range = 60%–100%).
Instructional Materials and Teaching Procedures
Phase 1: Instruction in the Math Classroom
EAI. The video anchor Fraction of the Costdepicts three students who decide to construct a skateboard ramp
rather than buy one. They find some two-by-fours in the garage that they can use for the project, but they must
calculate how much additional wood and other materials they need to build the ramp. To determine how much
money each student can contribute to the project, the students must add and subtract whole numbers and cal-
culate percentages.
To solve the problem, students accessed relevant information on the CD, which included the dimensions of
the skateboard ramp, the cost of materials, and the money available for buying the materials. Using this infor-
mation, the students attempted to determine (a) the amount of money they had to spend if they pooled it, (b)
the most efficient use of the wood they already had, (c) the materials they needed to build the ramp (lengths
and quantity of screws, lengths of two-by-fours, thickness of decking material), and (d) the most efficient use
of the wood they still needed to buy. Students received full credit for showing one of the two possible solu-
tions to the problem. Students could also receive partial credit if they provided a close but not totally accurate
solution. Within six 45-minute class sessions, most groups of students had arrived at one of the five full or
partial solutions.
The teachers facilitated class discussions by asking students questions such as, “What are some of the things
you need to know when solving the problem?” These classroom discussions helped keep the students on track
and reinforced their newly acquired understandings. Students could test their ideas by measuring two-by-fours
in the classroom that had been cut to the same lengths as the wood shown in the video. Students could also
inspect a small-scale model of the ramp to determine how the ramp frame was put together. The teachers
instructed the students to work in pairs. On the last day of Phase 1, the groups presented their solutions to the
rest of the class.
TBI. Students in TBI worked on solving word problems and constructed response problems that were not
posed in a video format. The teacher taught students an eight-step strategy for solving word problems: read,
paraphrase, visualize, (re)state, hypothesize, estimate, compute, and self-check (Montague, 1997). The teach-
ers encouraged the students to create a mnemonic device to help them remember the eight-step process.
Students created mnemonics such as “Robert picked very red hot eggplants cause he’s crazy” or “Remember
parents very realistically hope that every child can.” As the teacher displayed each word problem using an
overhead projector, students discussed how to solve the problem and then wrote their answers in their folders.
The following are examples of the word problems:
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(cid:129) Word problem 1. During the first week in January, Mary earned $43.00 at her job at Pizza Hut. She
spent $24.75 on clothes and $10.75 on entertainment. Her friend, Ann, bought her lunch at Burger
King for $3.25. How much money did Mary have left out of her first week’s paycheck?
(cid:129) Word problem 2. Roy wanted to build a model glider. The main wing needed to be 25 1/2 in. long.
The back wing had to be 6 3/4 in. long. How much wood was needed for both wings?
After students finished solving the word problems, they worked on the worksheets in their packets while they
waited for their classmates to finish the word problems. This format allowed the students to work at their own
pace. On average, the class managed to solve two to three problems a day.
Each day after students solved the word problems, the teacher gave them hands-on problems that comple-
mented the concepts taught in the word problems. Each problem engaged students for two or more class ses-
sions. The following are examples of the hands-on problems:
(cid:129) Applied problem 1. The purpose of this project was for students to work in pairs and build a picnic
table out of straws. Each student pair was given six straws that were to be assumed to measure 8 ft.
length. The students’finished table had to have a top and be strong enough to support an egg. Once
a pair of students had created their table, they had to measure the table’s dimensions, create blue-
prints, calculate the cost of their final product, and check its weight capacity.
(cid:129) Applied problem 2. The goal of this project was for students to build the cheapest, strongest, and
tallest structure measuring a minimum of 20 in. in length and 18 1/2 in. in height. Upon comple-
tion, this structure had to support a brick for 20 seconds. The students were given an imaginary
budget of $100 with which they could purchase paperclips, cards, and a fold in the card for $1
each.
The students in TBI worked on these applied problems and other word problems in their academic classroom
for six days.
Posttest scores and classroom observations showed that several EAI and TBI students with disabilities did not
understand the concepts that they were supposed to have learned in the math classroom. We decided to test
whether four of these students would profit from more individualized instruction on solving the video-based
problem. Two students from the EAI group and two students from the TBI group worked in pairs with the
special education teacher in the resource room for 50-minute sessions over five days. The intervention group
also served as a pilot group to test revisions and additions to the Fraction of the Costvideo, including a color-
coded, rotating ramp; an interactive, click-and-drag help module; and a talking calculator to help students
compute whole numbers. Each pair of students shared one laptop computer, which they used to search rele-
vant parts of the video. They also used the click-and-drag modules of the CD to help them visualize the best
way to use the wood.
Phase 2: Related Instruction in the Technology Education Classroom
In the technology education classroom, EAI and TBI students worked on building a hovercraft rollover cage
as economically as possible out of four 10 ft. lengths of PVC pipe and connectors. The central concepts in the
hovercraft project paralleled those taught in the math classrooms 6, 13, or 22 weeks before. Before they built
the cage, students worked on designing and building scale models of a house and a hovercraft frame out of
drinking straws. The students voted on the four best designs, which they then used to construct the full-size
hovercraft cage. Once the cages were complete, the students lifted them onto a 4 x 4 ft. plywood platform
(i.e., the hovercraft) and inserted a leaf blower into a hole in the plywood. The leaf blower elevated the hover-
craft slightly above the floor. Students took turns riding the hovercraft and competed in relays, racing from
one end of the technology education classroom to the other.
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Fidelity of Treatment Implementation
Several methods helped ensure that teachers followed the instructional plans. During Phase 1, one researcher
observed the EAI and TBI classrooms 100% of the time. On 10% of those occasions, a second observer was
present. Independently, the observers took notes on the classroom activities and later compared their notes,
identifying similarities and differences. Although no substantive discrepancies in instruction were noted,
minor inconsistencies in the amount of detail provided to students and in the pacing of lessons were brought
to the teachers’attention. One researcher attended all of the classes in the special education classroom. During
Phase 2, a researcher attended 81% of the class periods, with a second observer also on hand 10% of the time.
Several class periods were also videotaped, which provided a readily available look at what transpired during
all three phases of the study.
Results
We describe the results in two ways to provide the most complete picture of the findings. In the first section
below, we present statistical analyses of the test performance of all students (with and without disabilities)
who participated in the learning activities in the mathematics and technology education classrooms. In the
second section, we provide a more detailed description of how students with disabilities fared on the math
problems in the mathematics, special education, and technology education settings.
Performance of Students with and without Disabilities
Math classroom. Table 2 reports the obtained and adjusted means and standard deviations of the participating
students. We conducted 2 x 2 analyses of covariance (ANCOVA) on the fractions computation, word problem,
and video problem posttests, with each pretest serving as the covariate. One factor was the class (Ann’s or
Marj’s), and the other was the type of instruction (EAI or TBI). (The probability of Type 1 error was main-
tained at .05 for all subsequent analyses.)
Results were mixed across the three posttests. On the fractions computation test, there was a significant inter-
action between class and type of instruction, F(1, 77) = 4.14, p= .04, η2= .05, but there was no main effect
for class, F(1, 77) = .55, p= .46, η2= .01, or for instruction, F(1, 77) = 1.34, p= .25, η2= .02. Taking into
account differences in pretest scores, the adjusted posttest mean of Marj’s TBI class was 2 to 3 points higher
than those of the other three classes. On the word problem test, there was a main effect for type of instruction
in favor of TBI, F(1, 83) = 9.30, p= .003, η2= .10, but not for class, F(1, 83) = 1.43, p= .23, η2= .02, or for
class by type of instruction, F(1, 83) = .31, p= .58, η2= .00. On the video problem test, there was a main
effect for type of instruction in favor of EAI, F(1, 67) = 17.32, p= .000, η2= .21, but not for class, F(1, 67) =
.05, p= .83, η2= .00, or for class by type of instruction, F(1, 67) = .96, p= .33, η2= .01.
Technology education classroom. Table 3 shows the scores of students on the hovercraft problem-solving test
before and after instruction in the technology education classroom. To determine how much EAI and TBI stu-
dents remembered from instruction in the mathematics classroom, we conducted a 2 x 3 ANCOVA on the
hovercraft pretests, with the fractions computation, word problem, and video posttests serving as covariates.
One factor was type of instruction (EAI or TBI), and the other was technology education session (6, 13, or 22
weeks). The analysis showed a main effect for type of instruction in favor of the EAI group, F(1, 33) = 6.98, p
= .01, η2= .17, and for session, F(2, 33) = 10.32, p= .00, η2= .385, but not for type of instruction by ses-
sion, F(2, 33) = .289, p= .75, η2= .02.
We used a 2 x 3 ANCOVA to analyze the performance of students in each group (EAI or TBI) in each tech-
nology education session (6, 13, or 22 weeks). The tests showed differences in students’performance depend-
ing on the technology education session they attended, F(2, 36) = 3.75, p= .03, η2= .17. Students in the last
technology education cohort scored almost 3 points higher than students in either of the other two cohorts.
There were no differences for group, F(1, 36) = .31, p= .58, η2= .01, or for group by session, F(2, 36) = .37,
p= .70, η2= .02.
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Performance of Students with Disabilities
Because of the small sample size of students with disabilities in each group, statistical tests could not be con-
ducted. The combined posttest scores of students with disabilities in the EAI and TBI groups were about half
those of students without disabilities: 15.1 with disabilities, 29.7 without disabilities on computation test; 15.2
with disabilities, 27.6 without disabilities on word problem test; 7.9 with disabilities, 12.2 without disabilities
on video test. However, all four students who worked on the EAI problem in the special education setting
improved their posttest scores considerably over what they had earned the first time they took the video
posttest (2 pretest, 30 posttest, +28 points; 0 pretest, 21 posttest +21 points; 11 pretest, 34 posttest, +23
points; 9 pretest, 28 posttest, +19 points). In the following paragraphs, we describe in detail the work of two
students with disabilities (Will, Serena) whose performance is representative of the other two students with
disabilities who received small-group instruction on the video-based problem.
Will. Will had a pleasant disposition and was motivated to learn math. His individualized education program
(IEP) stated that he was a “slow learner” with learning disabilities who needed special education services in
speech and language and in math. Each week, he received five hours of math instruction and one hour of
speech instruction in the regular or special education classroom.
Will watched the video-based problem in the math classroom with interest. The video shows three friends—
Cindy, Ryan, and Michael—deciding that they should each contribute the same amount money to the skate-
board ramp project. Cindy has $19 left over from her birthday money and is allowed to spend all of it. Ryan
has $210 in his savings account and is allowed to withdraw 10% at a time. Michael has $73 in his savings
account and must keep a minimum balance of $50. Together, the friends have a total of $63. This amount
divided by 3 is $21, $2 more than Cindy can afford. Because Cindy has only $19, the total amount the friends
can spend on the ramp is $57.
Will understood that Cindy could spend $19 and that Michael could spend $23 ($73 - $50), but he did not
know how to calculate 10% of Ryan’s savings account balance. He wandered to a neighboring group to find
out how much Ryan could spend. When he obtained this information, he went back to his own group and tried
to convince his partners that the three friends could spend $57. On subsequent days, Will searched portions of
the video for more relevant information but also continued to ask students in other groups for answers to the
other subproblems.
During two days in the math classroom, the special education teacher remarked that Will was “extremely
engaged.” She added that “he likes the video, measuring, and the ramp. It is the tactile stuff that he really
likes.” Although Will worked diligently on parts of the problem, he did not contribute much to whole-class
discussions. Notes from classroom observations revealed that he became frustrated when he was unable to add
mixed numbers (lengths of wood available for building the ramp). Will improved his computation scores from
4 on the pretest to 15 on the posttest, but he showed little or no new understanding of the video problem
(pretest = 0, posttest = 2).
In the resource room, Will worked with another student with disabilities, Serena, to solve the video problem.
Will was excited about getting the chance to work on the computer. He remembered from the math classroom
that each person in the video had contributed $19 to the project. He seemed proud of himself after computing
10% of $210 with a calculator and adding each person’s contribution ($19 + $19 + $19 = $57) without help.
However, Will was confused as he tried to find the dimensions of each part of the ramp shown in the video
that corresponded to the dimensions of lumber in the schematic drawing. On the next day, he used the color-
coded click-and-drag module to figure out which lengths of wood on the CD corresponded to the dimensions
in the plan.
On the final day of the intervention in the resource room, Will and his partner calculated the total cost of
building the skateboard ramp. Unfortunately, Will did not get the chance to construct the hovercraft cage in
the technology education classroom because his family moved out of the school district.
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Serena. Serena was labeled “other health impaired” (OHI), did not like school, and seemed unhappy much of
the time. One day, her math teacher asked why she had not come to school the day before. Serena answered
“... because I did not feel like or want to come to school, so I didn’t.” When the other students were engaged
in solving the TBI problems in the math classroom, Serena said very little and did not interact with the
teacher or the other students. In fact, whenever students came too close to her, Serena reacted violently, trying
to poke them with her pencil. When she was not staring blankly into space, she drew pictures of devils cov-
ered in blood.
In the resource room, Serena at first complained about having to solve the video problem. However, she and
her partner, Will, gradually grew more interested in the problem when they realized that they had the comput-
er all to themselves. Serena took charge of the mouse and told Will that “we should write something down so
we can buy something.” She paused at several points in the video where she thought important information
was located, such as the store ad that listed the cost of lumber, screws, and other materials. At one point, she
told Will that the students in the video needed to use treated lumber to build the ramp or else the ramp would
warp (one of the students in the video provided this fact). Serena used a calculator to compute costs and then
checked her findings with Will’s.
One day, Serena and Will got into a debate about how many two-by-fours the students in the video needed for
the ramp. Serena counted seven (the correct answer), but Will insisted they needed more. So Serena explained
to Will how she arrived at seven. When she could not convince him that they needed seven boards, she hit her
head with her hand in frustration. Together, Serena and Will eventually figured out that Will had mistakenly
added the two-by-fours in the garage to the two-by-fours they needed to buy.
In the resource room, Serena’s behavior changed drastically. She seemed happy and took charge of solving the
problem. For example, she teased Will about not copying prices of materials into their problem-solving book-
let as fast as she did. She told him that he was “as slow as heck.” As Serena became more engrossed in the
problem, she did not mind that Will’s arm almost touched hers. On only a few occasions did she point her
pencil at Will as if she were going to stab him.
In the technology education classroom, Serena returned to the behaviors she had exhibited in the math class-
room. At times, she seemed interested in the classroom activities, but she became withdrawn at other times.
When the technology education teacher asked students to build the largest house possible using plastic straws,
Serena was interested and engaged. In fact, she built the second-largest house in the class. She was surprised
by her accomplishment and visibly excited. However, this excitement faded as the students began to design
the hovercraft cage. Once again, she lost interest and did not communicate with her classmates or participate
in the planning.
Discussion
The purpose of this study was to compare the math achievement of students with and without disabilities in
several learning contexts and instructional conditions. Our results show that, overall, students in the mathe-
matics classroom learned with EAI or TBI. That is, EAI students scored higher on the video problem test than
TBI students and, conversely, TBI students scored higher on the word problem test than EAI students. These
results are not surprising because we would expect students to score better on well-designed, curriculum-
aligned measures. Despite the practice students gained in computing fractions while working on both kinds of
problems, no differences were found in computation scores of either group.
Several weeks later (6, 13, 22), students were tested in the technology education classroom on a transfer test
that was similar in concept but different in format than the ones they had worked on in their math classroom
(EAI or TBI). This transfer task was actually the pretest for an applied problem that the math, technology edu-
cation, and special education helped the researchers design. Results of the transfer test showed that EAI stu-
dents remembered the central concepts they had learned in the math classroom better than TBI students did.
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Posttests administered after instruction on the hovercraft construction showed that instruction in the technolo-
gy education classroom closed the achievement gap between EAI and TBI students.
Four students with disabilities who received extra help on the video-based problem in the special education
classroom improved their scores on the video test by almost 23 points. This was a remarkable improvement
considering they only spent four additional days working on it with the special education teacher in her
resource room. Their scores on the hovercraft tests indicated little or no transfer of concepts from the math to
the technology education classroom.
Limitations
Several limitations temper our findings. First, the school schedule forced us to use intact classes and did not
permit us to assign students randomly to instructional groups. Second, we were unable to analyze with statis-
tics the scores of students with disabilities because of the relatively small number of students in each compari-
son group. Third, absences from school and transfers to other schools reduced the number of students who
participated in all phases of study. We had anticipated and planned for this problem, but the attrition rate was
higher than we expected.
Practical Significance
In this study, mathematics, special education, and technology education teachers with diverse but complemen-
tary skills joined together to plan, develop, and deliver the central concepts embedded in the math and tech-
nology education curriculum. The impact of each teacher was clearly evident throughout the study. The math
teachers assisted students in interpreting and illuminating the math concepts embedded in the EAI and TBI
problems. When the special education teacher realized that some of the students with disabilities did not
understand how to solve the problems in the math classroom, she used her specialized skills to help them in
the resource room.
The technology education teacher translated the math concepts into a motivating project that asked students to
apply and practice what they had previously learned. In this study, the hovercraft project motivated students to
do academic work, in keeping with recommendations by the President’s Committee of Advisors on Science
and Technology (PCAST, 1997) and the International Society for Technology in Education (ISTE) National
Educational Technology Standards Project (ISTE, 1999) that recommended technology education focus on
learning, emphasizing content and pedagogy, not just hardware. In this study, students benefited from the
opportunity to learn using the video-based anchors in the math classroom and the hovercraft projects in the
technology education classrooms.
As for accommodating the needs of students with disabilities, this study showed that instruction in general edu-
cation classrooms without additional, individualized help was not sufficient to raise their academic perform-
ance. However, the four students who received additional instruction from a skilled special education teacher
scored well above the average of their peers without disabilities on the video posttest. For a variety of philo-
sophical, legal, social, budgetary, and educational reasons, special educators are advocating for students with
disabilities to be educated alongside their peers without disabilities in general education classrooms. Although
some authors have found that the performance of students with disabilities in inclusive classrooms equals or
surpasses their performance in pullout settings (e.g., Rea, McLaughlin, & Walther-Thomas, 2002), the results of
this study support other authors who warn that even in general education classes with small groups of students,
some students with disabilities do not get the individualized attention they need to understand the academic
content (Baxter, Woodward, Voorhies, & Wong, 2002, Bottge et al., 2002; Kauffman, 1999).
The goal of designing and implementing instructional materials and plans across subject areas was, for the
most part, achieved in this study. Students without disabilities profited from the contextualized instruction in
their math classroom as demonstrated by their ability to transfer their skills across content areas. However, the
results also showed that students with disabilities required more attention than both the general education and
special education teacher could give in the whole-class setting. When the instruction was individualized to the
needs of students with disabilities, they were able to profit from the integrated curriculum.
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